Suppose that the vector of SNP j’s effects on the traits βj is drawn from a mixture of mean-zero multivariate normal distributions. The distribution of component c=1,2,…,C is βj|c~N(0,Ωc), and its mixture weight is denoted pc, where ∑c=1Cpc=1. In this case, the z-statistic associated with the MTAG estimate β^MTAG,j,t is a mixture distribution with component distributions Zj,t|c~N(0,ωt′ωtt(Ω−ωtωt′ωtt+∑j)−1(Ωc+∑j)(Ω−ωtωt′ωtt+∑j)−1ωtωttωt′ωtt(Ω−ωtωt′ωtt+∑j)−1ωtωtt).To define power and FDR, let D denote the set of components such that a SNP is null for trait t (i.e., the tth element of βj is drawn from a degenerate distribution with all mass on 0). Power for trait t can be calculated as Power≡Pr(|Zj,t|>z0|c∉D)=∑c∉DPr(|Zj,t|>z0|c)pc∑c∉Dpc,where z0 is the z-statistic associated with genome-wide significance. The FDR for trait t can be calculated as FDR≡Pr(null ||Zj,t|〉z0)=Pr(|Zj,t|>z0 | null)Pr(null)Pr(|Zj,t|>z0)=∑c∈DPr(|Zj,t|>z0|c)pc∑c=1CPr(|Zj,t|>z0|c)pc.As with the MSE formula, we verify in simulations that these formulas are good approximations when using estimates of Ω and ∑j (Supplementary Table 1).
